IT3_2005-6_paper_v2

# IT3_2005-6_paper_v2 - INFORMATION THEORY 3 MATH 34600 FINAL...

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INFORMATION THEORY 3 MATH 34600 FINAL EXAM PAPER 2005/06 This paper contains three questions; each is worth 25 marks. A candidate’s two best answers will be used for assessment. Time given is 2(?) hours. Date of the exam: dd/mm/2005 ... am to ... pm. A calculator of the approved type is allowed. 1

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1. Consider the source X with alphabet A = { α, β, γ, δ, ε, ϕ } and probability distribution P X : x α β γ δ ε ϕ P X ( x ) .3 .2 .15 .125 .125 .1 (a) State Kraft’s condition for the existence of a prefix-free binary code for a general alphabet X with supposed codeword lengths x (inte- gers), x ∈ X . [4 marks] (b) Construct the Shannon-Fano code for the source X , writing down the required code word lengths first. [6 marks] (c) Calculate the expected length of your code, and the entropy of the source X . [4 marks] (d) Explain the significance of the entropy in the context of Shannon’s source coding theorem. Be careful to define any notation and termi- nology you introduce. [3 marks] (e) Still referring to Shannon’s source coding theorem: Explain in words the difference of performance of your code constructed in (b) and the asymptotically optimal source code. [3 marks] (f) Can you improve the expected length of the code you constructed in part (b)? If so, show how to do this. (Hint: is the Kraft inequality
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